Cryptography is the study of mathematical techniques that provide the base of secure communication in the presence of malicious adversaries. The main goals of secure communication include confidentiality of data, integrity of data and authentication of entities involved in a transaction. Historically, “symmetric key” cryptography was used to attempt to meet the goals of secure communication. However, symmetric key cryptography requires entities to exchange secret keys through a secret channel prior to communication. One weakness of symmetric key cryptography is the security of the secret channel. Public key cryptography provides a means of securing a communication between two entities without requiring the two entities to exchange secret keys through a secret channel prior to the communication. An example entity “A” selects a pair of keys: a private key that is only known to entity A and is kept secret; and a public key that is known to the public. If an example entity “B” would like to send a secure message to entity A, then entity B needs to obtain an authentic copy of entity A's public key. Entity B encrypts a message intended for entity A by using entity A's public key. Accordingly, only entity A can decrypt the message from entity B.
For secure communication, it is essential that entity A select the pair of keys such that it is computationally infeasible to compute the private key given knowledge of the public key. This condition is achieved by the difficulty (technically known as “hardness”) of known mathematical problems such as the known integer factorization mathematical problem, on which is based the known RSA algorithm, which was publicly described in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman.
Elliptic curve cryptography is an approach to public key cryptography based on the algebraic structure of elliptic curves over finite mathematical fields. An elliptic curve over a finite field, K, may be defined by a Weierstrass equation of the formy2+a1xy+a3y=x3+a2x2+a4x+a6.  (0.1)If K=, where p is greater than three and is a prime, equation (0.1) can be simplified toy2=x3+ax+b.  (0.2)If K=, i.e., the elliptic curve is defined over a binary field, equation (0.1) can be simplified toy2+xy=x3+ax2+b.  (0.3)
The set of points on such a curve (i.e., all solutions of the equation together with a point at infinity) can be shown to form an abelian group (with the point at infinity as the identity element). If the coordinates x and y are chosen from a large finite field, the solutions form a finite abelian group.
Elliptic curves cryptosystems rely on the hardness of a problem called the elliptic curve discrete logarithm problem (ECDLP). Where P is a point on an elliptic curve E and where the coordinates of P belong to a finite field, the scalar multiplication kP, where k is a secret integer, gives a point Q equivalent to adding the point P to itself k times. It is computationally infeasible, for large finite fields, to compute k knowing P and Q. The ECDLP is: find k given P and Q (=kP).